A random multiplicative function is a multiplicative function alpha(n) such that its values on primes, (alpha(p))_(p=2,3,5,...), are i.i.d. random variables. The simplest example is the Steinhaus function, which is a completely multiplicative function with alpha(p) uniformly distributed on the unit circle. A basic question in the field is finding the limiting distribution of the (normalized) sum of alpha(n) from n=1 to n=x, possibly restricted to a subset of integers of interest. This question is currently resolved only in a few cases. We shall describe ongoing work where we are able to find the limiting distribution in many new instances of interest. The distribution we find is not gaussian, in contrast to all previous works. This is joint work with Mo Dick Wong (Durham University).